%0 Journal Article
%T About the algebraic closure of the field of power series in several variables in characteristic zero
%+ Institut de MathÃ©matiques de Marseille (I2M)
%A Rond, Guillaume
%Z 51 pages
%< avec comitÃ© de lecture
%@ 1949-2006
%J Journal of Singularities
%I Worldwide Center of Mathematics, LLC
%V 16
%P 1-51
%8 2017-12
%D 2017
%Z 1303.1921
%R 10.5427/jsing.2017.16a
%K Abhyankar-Jung theorem
%K Eisenstein theorem
%K Abhyankar valuation
%K Diophantine approximation
%Z 13F25 (11J25, 12J20, 12F99, 13J05, 14B05, 32B10)
%Z Mathematics [math]/Commutative Algebra [math.AC]
%Z Mathematics [math]/Algebraic Geometry [math.AG]
%Z Mathematics [math]/Complex Variables [math.CV]Journal articles
%X We begin this paper by constructing different algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice of an Abhyankar valuation and is constructed via a generalization of the Newton-Puiseux method for this valuation. Then we study the Galois group of a polynomial with power series coefficients. In particular by examining more carefully the case of monomial valuations we are able to give several results concerning the Galois group of a polynomial whose discriminant is a weighted homogeneous polynomial times a unit. One of our main results is a generalization of Abhyankar-Jung Theorem for such polynomials, classical Abhyankar-Jung Theorem being devoted to polynomials whose discriminant is a monomial times a unit.
%G English
%L hal-01313076
%U https://hal.archives-ouvertes.fr/hal-01313076
%~ CNRS
%~ EC-MARSEILLE
%~ INSMI
%~ I2M-2014-
%~ I2M
%~ UNIV-AMU
%~ TEST-AMU